Prove Invertibility of Matrix I + A for Projection Matrices A

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Homework Help Overview

The discussion revolves around proving the invertibility of the matrix I + A, where A is a projection matrix satisfying the condition A² = A. Participants are exploring the properties of projection matrices and their implications for matrix operations.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants are considering various methods to demonstrate the invertibility of I + A, including testing specific forms of potential inverses and verifying their products. Questions arise regarding the validity of their approaches and the implications of matrix multiplication.

Discussion Status

There is an ongoing exploration of the relationship between the matrices I + A and I - A/2, with some participants confirming that their product yields the identity matrix. This leads to a discussion about the definition of an inverse and whether this confirms the invertibility of I + A.

Contextual Notes

Participants are working under the assumption that A is a projection matrix, which influences their reasoning about the properties of I + A. There is an emphasis on verifying calculations and understanding the implications of matrix multiplication in the context of inverses.

danik_ejik
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Hello,
I need a hint on how to begin this proof, please.

Prove that if A is a projection matrix, A2=A, then I + A is invertible and
(I + A) -1 = I - \frac{1}{2}A.
 
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I would write B=a*I+b*A and compute B*(I+A) and (I+A)B and see for what values of a and b I find an inverse. Note that A^2=A, so there are no higher powers in the series expansion of the inverse.
 
If I+A and I-A/2 are supposed to be inverses then their product should be I, right? Is it?
 
Well, I-A/2 is a matrix. Why don't you try to see what you get if you multiply it by I+A? Maybe it will tell you something?
 
I tried multiplying (I + A) and (I - A/2) and indeed it equals I. That's it!? That proves it ?
 
danik_ejik said:
I tried multiplying (I + A) and (I - A/2) and indeed it equals I. That's it!? That proves it ?

Sure that proves it. If MN=I then M=N^(-1) and N=M^(-1). It's the definition of 'inverse'.
 
thank you
 

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